Existence of global strong solution and vanishing capillarity-viscosity ...
Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system Fr´ed´eric Charve∗, Boris Haspot
†‡
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Abstract In the first part of this paper, we prove the existence of global strong solution for Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the Korteweg system converges in the case of a γ law for the pressure (P (ρ) = aργ , γ > 1) to entropic solution of the compressible Euler equations. In particular it justifies that the Korteweg system is suitable for selecting the physical solutions in the case where the Euler system is strictly hyperbolic. The problem remains open for a Van der Waals pressure because in this case the system is not strictly hyperbolic and in Lax,G particular the classical theory of Lax and Glimm (see [21, 11]) can not be used.
1
Introduction
We are concerned with compressible fluids endowed with internal capillarity. The model we consider originates from the XIXth century work by Van der Waals and Korteweg VW,fK [38, 22] and was actuallyfDS,fJL,fTN derived in its modern form in the 1980s using the second gradient theory, see for instance [9, 20, 37]. The first investigations begin with the Young-Laplace theory which claims that the phases are separated by a hypersurface and that the jump in the pressure across the hypersurface is proportional to the curvature of the hypersurface. The main difficulty consists in describing the location and the movement of the interfaces. Another major problem is to understand whether the interface behaves as a discontinuity in the state space (sharp interface) or whether the phase boundary corresponds to a more regular transition (diffuse interface, DI). The diffuse interface models have the advantage to consider only one set of equations in a single spatial domain (the density takes into account the different phases) which considerably simplifies the mathematical and numerical study (indeed in the case of sharp interfaces, we have to treat a problem with free boundary). ∗
Universit´e Paris-Est Cr´eteil, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees (UMR 8050), 61 Avenue du G´en´eral de Gaulle, 94 010 Cr´eteil Cedex (France). E-mail: [email protected] † Basque Center of Applied Mathematics, Bizkaia Technology Park, Building 500, E-48160, Derio (Spain) ‡ Ceremade UMR CNRS 7534 Universit´e de Paris IX- Dauphine, Place du Marchal DeLattre De Tassigny 75775 PARIS CEDEX 16 , [email protected]
1
Let us consider a fluid of density ρ ≥ 0, velocity field u ∈ R, we are now interested in the following compressible capillary fluid model, which can be derived from a fDS Cahn-Hilliard like free energy (see the pioneering work by J.E. Dunn and J. Serrin in [9] and also in fA,fC,fGP,HM [1, 3, 12, 17]). The conservation of mass and of momentum write: ∂ ρ + ∂x (ρ u ) = 0, ∂t (1.1) ∂ (ρ u ) + ∂x (ρ (u )2 ) − ∂x (ρ ∂x u ) + ∂x (a(ρ )γ ) = 2 ∂x K, ∂t
3systeme
where the Korteweg tensor reads as following:
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1 0 divK = ∂x ρ κ(ρ )∂xx ρ + (κ(ρ ) + ρ κ (ρ ))|∂x ρ |2 − ∂x κ(ρ )(∂x ρ )2 . 2
(1.2)
κ is the coefficient of capillarity and is a regular function of the form κ(ρ) = 2 ρα with 2 α ∈ R. In the sequel we shall assume that κ(ρ) = ρ . The term ∂x K allows to describe the variation of density at the interfaces between two phases, generally a mixture liquidvapor. P = aργ with γ ≥ 1 is a general γ law pressure term. corresponds to the controlling parameter on the amplitude of the viscosity and of the capillarity.Hprepa When we 3systeme set v = u + ∂x (ln ρ ), we can write (1.1) on the following form (we refer to [13] for the computations): ∂ ρ + ∂x (ρ v ) − ∂xx ρ = 0, ∂t (1.3) ∂ (ρ v ) + ∂x (ρ (u )(v )) − ∂x (ρ ∂x v ) + ∂x (a(ρ )γ ) = 0, ∂t
divK
1.1
1.1
We now consider the Cauchy problem of (1.3) when the fluid is away from vacuum. 1.1 Namely, we shall study (1.3) with the following initial data: ρ (0, x) = ρ0 (x) > 0, u (0, x) = u0 (x),
(1.4)
1.2
such that: lim
x→+,−∞
(ρ0 (x), u0 (x)) = (ρ+,− , u+,− ), with ρ+,− > 0. 1.1
We would like to study in the sequel the limit process of system (1.3) when goes to 0 and to prove in particular that we obtain entropic solutions of the Euler system: ∂ ρ + ∂x (ρv) = 0, ∂t (1.5) ∂ (ρv) + ∂x (ρv 2 ) + ∂x (aργ ) = 0, ∂t Let us now explain the interest of the capillary solutions for the hyperbolic systems of conservation laws.
1.1
Viscosity capillarity processes of selection for the Euler system
In addition of modeling a liquid-vapour mixture, the Korteweg also shows purely theoretical interests consisting in the selection of the physically relevant solutions of the 2
1.3
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Euler model (in particular when the system is not strictly hyperbolic). The typical case corresponds to a Van der Waals pressure: indeed in this case the system is not strictly hyperbolic in the elliptic region (which corresponds to the region where the phase change occurs). In the adiabatic pressure framework (P (ρ) = ργ with γ > 1), the system is strictly hyperbolic and the theory is classical. More precisely we are able to solve the Riemann problem when the initial Heaviside data is small in the BV space. Indeed we are in the context of the well known Lax result as the system is also genuinely nonlinear (we refer Lax to [21]). It means we have existence of global C 1 -piecewise solutions which are unique in the class of the entropic solutions. This result as been extent by Glimm in the context of small initial data in the BV-space by using a numerical scheme and approximating the initial BV data by a C 1 -piecewise function (which implies to locally solve the Riemann problem via the Lax result). For BB1 the uniqueness of the solution we refer to the work of Bianchini and Bressan ([2]) who use a viscosity method. In the setting of the Van der Waals pressure, the existence of global solutions and the nature of physical relevant solutions remain completely open. Indeed the system is not strictly hyperbolic anymore. If we rewrite the compressible Euler system in Lagrangian coordinates by using the specific volume τ = 1/ρ in ( 1b , ∞) and the velocity u, the system satisfies in (0, +∞) × R the equations: ∂t τ − ∂x u = 0, (1.6) ∂t u − ∂x (Pe(τ )) = 0,
euler
with the function Pe : ( 1b , ∞) → (0, ∞) given by: 1 Pe(τ ) = P ( ), τ The two eigenvalues of the system are: q λ1 (τ, v) = − −Pe0 (τ ),
1 τ ∈ ( , ∞). b
q λ2 (τ, v) = − −Pe0 (τ ).
The corresponding eigenvectors r1 , r2 are: ! 1 q w1 (τ, v) = , w2 (τ, v) = −Pe0 (τ )
q 1 − −Pe0 (τ )
(1.7)
! (1.8)
Furthermore by calculus we obtain: 00 00 −Pe (τ ) Pe (τ ) ∇λ1 (τ, v) · w1 (τ, v) = q , ∇λ2 (τ, v) · w2 (τ, v) = q 2 −Pe0 (τ ) 2 −Pe0 (τ )
(1.9)
We now recall the definition of a standard conservation law in the sense of Lax (it means entropy solutions): • The system is strictly entropic if the eigenvalues are distinct and real. 3
vp
• The characteristics fields are genuinely nonlinear if we have for all (τ, v), ∇λ1 (τ, v) · w1 (τ, v) 6= 0 and ∇λ2 (τ, v) · w2 (τ, v) 6= 0, Serre
for more details we refer to [33]. The definition of genuine nonlinearity is some kind of extension of the notion of convexity to vector-valued functions (in particular when we consider the specific case of the traveling waves). The previous assumptions aim at Evans Serre ensuring the existence and the uniqueness of the Riemann problem ( see [10] and [33]). When P is a Van der Waals pressure, we observe that the first conservation law Serre,Evans ([33, 10]) is far from being a standard hyperbolic system, indeed: • It is not hyperbolic (but elliptic) in ( α11 , α12 ) × R,
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• the characteristic fields are not genuinely nonlinear in the hyperbolic part of the state space. Here the classical Lax-Glimm theory cannot be applied. In particular there doesn’t exist any entropy-flux pair, which suggests that the entropy framework is not adapted for selecting the physically relevant solutions. In order to deal with this problem, Van der Waals and Korteweg began by considering the stationary problem with null velocity, and Rohdehdr solving ∇P (ρ) = 0. For more details we refer to [31]. It consists in minimizing in the following admissible set Z 1 1 A0 = {ρ ∈ L (Ω)/W (ρ) ∈ L (Ω), ρ(x)dx = m}, Ω
the following functionnal Z F [ρ] =
W (ρ(x))dx. Ω
Unfortunately this minimization problem has an infinity of solutions, and many of them are physically irrelevant. In order to overcome this difficulty, Van der Waals in the XIXth century was the first to regularize the previous functional by adding a quadratic term in the density gradient. More precisely he considered the following functional: Z 2 Flocal = W (ρ (x)) + γ |∇ρ |2 dx, 2 Ω with: Alocal = H 1 (Ω) ∩ A0 . This variational problem has a unique solution and its limit (as goes to zero) converge to a physical solution of the equilibrium problem for the Euler system with Van der Waals REF pressure, that was proved by Modica in [28] with the use of gamma-convergence. By the Euler-Lagrange principle, the minimization of the Van der Waals functional consists in solving the following stationary problem: ∇P (ρ ) = γ2 ρ ∇∆ρ , where the right-hand side can be expressed as the divergence of the capillarity tensor. Heuristically, we also hope that the process of vanishing capillarity-viscosity limit selects the physical relevant solutions as it does for the stationary system. This problem actually remains open. 4
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1.2
Existence of global entropic solutions for Euler system
Before presenting the results of this paper let us recall the results on this topic in these last decades. We shall focus on the case of a γ pressure law P (ρ) = aργ with γ > 1 and a positive. Let us mention that these cases are the only ones well-known (essentially because the system is strictly hyperbolic in this case and that we can exhibit many entropy-flux pairs). Here the Lax-Glimm theory can be applied, however at the end of the 70’s, one was interested in relaxing the conditions on the initial data by only assuming ρ0 and u0 in L∞ . In the beginning of the 80’s Di Perna initiated this program, consisting in obtaining global entropic solutions for L∞ initial data. Di1, Di2 1.3 Indeed in [7, 8], Di Perna prove the existence of global weak entropy solution of (1.5) 2 3 for γ = 1 + 2d+1 and γ = 2k + 2k + 1 (with k ≥ Ta 1), d ≥ 2 by using the so-called ”compensated compactness” introduced by Tartar in 35 [35]. This result was extended36by Chen Chen in [4] in the case γ ∈ (1, 35 ] and by Lions et al in [26] in the case γ ∈ [3, ∞). In [25], Lions et al generalize this result to the general case γ ∈ (1, 3), and finally the case γ = 1 Hu1 is treated by [18]. We would like to mention that these results are obtained through a vanishing artificial viscosity on both density and velocity. The problem of vanishing physical viscosity limit of compressible Navier-Stokes equations to compressible10Euler equations was until recently an open problem. However Chen and Perepelista in [5] proved that the solutions of the compressible Navier-Stokes system with constant viscosity coefficients converge to aHu2 entropic solution of the Euler system with finite energy. This result was extended in [19] to the case of viscosity coefficients depending on the density. 10 Hu2 Inspired by [5] and [19], we would like to show that the solution of the Korteweg system 1.1 (1.3) converges to a entropic solution of the Euler system with finite energy when the pressure is a γ law. To do this, we will prove for the first time up our knowledge the existence of global strong solution for the Korteweg system in one dimension in the case of Saint-Venant viscosity coefficients. By contrast, the problem of global strong solutions for compressible Navier-Stokes equations remains open (indeed one of the main difficulties consists in controlling the vacuum). This result justifies that the Korteweg system allows us to select the relevant physical solutions of the compressible Euler system at least when the pressure is adiabatic (P (ρ) = aργ with γ > 1). The problem remains open in the case of a Van der Waals pressure.
1.3
Results
Let us now describe our main result. In the1.1 first theorem we prove the existence of global strong solution for the Korteweg system (1.3). Theorem 1.1 Let ρ¯ > 0. Assume that the initial data ρ0 and u0 satisfy: 0 < m0 ≤ ρ0 ≤ M0 < +∞, ρ0 − ρ¯ ∈ H 1 (R), v0 ∈ H 1 (R) ∩ L∞ (R). 1.1
(1.10)
Then there exists a global strong solution (ρ, v) of (1.3) on R+ × R such that for every T > 0: ρ − ρ¯ ∈ L∞ (0, T, H 1 (R)), ρ ∈ L∞ (0, T, L∞ (R)), v ∈ L∞ (0, T, H 1 (R)) ∩ L2 (0, T, H 2 (R)) and v ∈ L∞ (0, T, L∞ (R)). 5
2.5
theo
Finally this solution is unique in the class of weak solutions satisfying the usual energy inequality. Remark 1 We would like to point out that the problem remains open in the case of the 1.1 Saint-Venant system, which corresponds to system (1.3) without capillarity. In the following theorem, we are interested in proving the convergence of the global 1.1 1.3 solutions of system (1.3) to entropic solutions of the Euler system (1.5).
theo1
Theorem1.1 1.2 Let γ > 35 and (ρ , v ) with m = ρ v theo be the global solution of the Cauchy problem (1.3) with initial data (ρ0 , v0 ) as in theorem (1.1).Then, when → 0, there exists a subsequence of (ρ , m ) that converge almost everywhere to a finite entropy solution 1.3 (ρ, ρv) to the Cauchy problem (1.5) with initial data (ρ0 , ρ0 v0 ). 36
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Remark 1 We would like to point out that Lions et al in [25] had obtained the existence of global entropic solution for γ > 1 by a viscosity vanishing process, and the considered regularizing system was exactly the Korteweg system modulo the introduction of the effective velocity. theo1
One important10 basis of our problem for theorem 1.2 is the following compactness theorem established in [5]. 10
Theorem 1.3 (Chen-Perepelitsa [5]) Let ψ ∈ C02 (R), (η ψ , q ψ ) be a weak entropy pair generated by ψ. Assume that the sequences (ρ (x, t), v (x, t)) defined on R × R+ with m = ρ v , satisfies the following conditions: 1. For any −∞ < a < b < +∞ and all t > 0, it holds that: Z tZ b (ρ )γ+1 dxdτ ≤ C(t, a, b), 0
(1.11)
1.8
(1.12)
1.9
(1.13)
1.10
a
where C(t) > 0 is independent of . 2. For any compact set K ⊂ R, it holds that Z tZ (ρ )γ+θ + ρ |v |3 dxdτ ≤ C(t, K), 0
K
where C(t, K) > 0 is independent of . 3. The sequence of entropy dissipation measures −1 η ψ (ρ , m )t + q ψ (ρ , m )x are compact in Hloc (R2+ ).
Then there is a subsequence of (ρ , m ) (still denoted (ρ , m )) and a pair of measurable functions (ρ, m) such that: (ρ , m ) → (ρ, m), a.e as → 0.
(1.14)
theo2 1.9
Remark 2 We would like to recall that the estimate (1.12) was firstPer derived by Lions et 35 al in [26] by relying the moment lemma introduced by Perthame in [30]. section2
The paper is arranged as follows. In section 2 we recall some important results on the notion of entropy enrtopy-flux pair for Euler system and on the kinetic formulation of 35 section3 theo section4 Lions et al in [26]. In section 3, we show theorem 1.1 and in the last section 4.1 we prove theo1 theorem 1.2. 6
1.11
2
Mathematical tools
section2
Definition 2.1 A pair of functions (η(ρ, v), H(ρ, v)) or (η(ρ, m), q(ρ, m)) for m = ρv, 1.1 is called an entropy-entropy flux pair of system (1.3), if the following holds: [η(ρ, v)]t + [H(ρ, v)]x = 0, 1.3
for any smooth solution of (1.5). Furthermore (η(ρ, v) is called a weak entropy if: η(0, u) = 0, for any fixed v. Definition 2.2 An entropy η(ρ, m) is convex if the Hessian ∇2 η(ρ, m) is nonnegative definite in the region under consideration.
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Such η satisfy the wave equation: ∂tt η = θ2 ργ−3 ∂xx η. 35
From [26], we obtain an explicit representation of any weak entropy (η, q) under the following form: Z η ψ (ρ, m) = χ(ρ, s − v)ψ(s)ds, R Z (2.15) ψ H (ρ, m) = (θs + (1 − θ)u)χ(ρ, s − v)ψ(s)ds,
1.4
R
where the kernel χ is defined as follows: χ(ρ, v) = [ρ2θ − v 2 ]λ+ , λ = and here:
3−γ 1 γ−1 > − , and θ = , 2(γ − 1) 2 2
tλ+ =tλ for t > 0, =0 for t ≤ 0, 35
Proposition 2.1 (see [26]) For instance, when ψ(s) = 12 s2 , the entropy pair is the mechanical energy and the associated flux: m2 m3 0 η ∗ (ρ, m) = + e(ρ), q ∗ (ρ, m) = 2 + e (ρ), (2.16) 2ρ 2ρ where e(ρ) =
κ γ γ−1 ρ
1.5
represents the gas internal energy in physics.
In the sequel we will work far away of the vacuum that it why we shall introduce equilibrium states such that we avoid the vacuum. Let (¯ ρ(x), v¯(x)) be a pair of smooth −,+ ) when − + x ≥ L for some monotone functions satisfying (¯ ρ(x), v¯(x)) = (ρ−,+ , v 0 1.1 large L0 > 0. The total mechanical energy for (1.3) in R with respect to the pair of reference function (¯ ρ(x), v¯(x)) is: Z 1 ρ(t, x)|v(t, x) − v¯(x)|2 + e∗ (ρ(t, x), ρ¯(x)) dx (2.17) E[ρ, v](t) = R 2 7
1.7
3system
0
where e∗ (ρ, ρ¯) = e(ρ)e (¯ ρ) − e (¯ ρ(ρ − ρ¯) ≥ 0. The total mechanical energy for system (??) κ with κ(ρ) = ρ is: Z E1 [ρ, u](t) = R
1 1 ρ(t, x)|u(t, x) − u ¯(x)|2 + e∗ (ρ(t, x), ρ¯(x)) + 2 (∂x ρ 2 )2 dx 2
(2.18)
1.7
(2.19)
1.7
1.1
and the total mechanical energy for system (1.3) is: Z 1 E2 [ρ, v](t) = ρ(t, x)|v(t, x) − v¯(x)|2 + e∗ (ρ(t, x), ρ¯(x)) dx R 2
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Definition 2.3 Let (ρ0 , v0 ) be given initial data with finite-energy with respect to the end states: (ρ± , v ± ) at infinity, and E[ρ0 , v0 ] ≤ E0 < +∞. A pair of measurable functions 1.3 (ρ, u) : R2+ → R2+ is called a finite-energy entropy solution of the Cauchy problem (1.5) if the following properties hold: 1. The total energy is bounded in time such that there exists a bounded function C(E, t), defined on R+ × R+ and continuous in t for each E ∈ R+ with for a.e t > 0: E[ρ, v](t) ≤ C(E0 , t). 2. The entropy inequality: η ψ (ρ, v)t + q ψ (ρ, v)x ≤ 0, is satisfied in the sense of distributions for all test functions ψ(s) ∈ {±1, ±s, s2 }. 3. The initial data (ρ0 , v0 ) are obtained in the sense of distributions. 1.2
10
We now give our main conditions on the initial data (1.4), which is inspired from [5]. Definition 2.4 Let (¯ ρ(x), v¯(x)) be some pair of smooth monotone functions satisfying (¯ ρ(x), v¯(x)) = (ρ−,+ , v −,+ ) when − + x ≥ L0 for some large L0 > 0. For positive constant C0 , C1 and C2 independent of , we say that the initial data (ρ0 , v0 ) satisfy the condition H if they verify the following properties: R • ρ0 > 0, R ρ0 (x)|u0 (x) − u ¯(x)| ≤ C0 < +∞, • The energy is finite: Z 1 ρ0 (x)|v0 (x) − v¯(x)|2 + e∗ (ρ0 (x), ρ¯(x)) dx ≤ C1 < +∞, R 2 • 2
Z R
|∂x ρ0 (x)|2 dx ≤ C2 < +∞. ρ0 (x)3−2α
In this section, we would like to 1.3 recall some properties on the pair of entropy for the 1.3 system (1.5). Smooth solutions of (1.5) satisfy the conservation laws: ∂t η(ρ, u) + ∂x H(ρ, u) = 0,
8
if and only if:
0
ηρρ
P (ρ) = ηuu . ρ2
(2.20)
ondes
(2.21)
initial
ondes
We supplement the equation 2.20 by giving initial conditions: η(0, u) = 0, ηρ (0, u) = ψ(u).
35
We are now going to give a sequel of proposition on the properties of η, we refer to [26] for more details. proputile
Proposition 2.2 For ρ ≥ 0, u, ω ∈ R, ondes
initial
• The fundamental solution of (2.20)-(2.21) is the solution corresponding to ηρ (0, u) = δ(u) is given by:
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χ(ρ, ω) = (ργ−1 − ω 2 )λ+ with λ = ondes
3−γ . 2(γ − 1)
(2.22)
initial
• The solution of (2.20)-(2.21) is given by: Z ψ(ξ)χ(ρ, ξ − u)dξ, η(ρ, u) =
(2.23)
entropie
(2.24)
flux
R
• η is convex in (ρ, ρu) for all ρ, u if and only if g is convex. • The entropy flux H associated with η is given by: Z γ−1 H(ρ, u) = ψ(ξ)[θξ + (1 − θ)ξ]χ(ρ, ξ − u)dξ where θ = . 2 R 35
We now give a important result on the entropy pair (see [26], lemma 4) . pair35
Proposition 2.3 Taking ψ(s) = 12 s|s|, then there exists a positive constant C > 0, depending only on γ > 1, such that the entropy pair (η ψ , H ψ ) satisfies: |η ψ (ρ, u)| ≤ (ρ|u|2 + ργ ), H ψ (ρ, u) ≥ C −1 (ρ|u|3 + ργ+θ ), for all ρ ≥ 0 and u ∈ R, ψ |ηm (ρ, u)| ≤ (ρ|u| + ρθ ),
(2.25)
2.37
ψ |ηmm (ρ, u)| ≤ Cρ−1 .
We are now going to give recent results on the entropy pair (η ψ , q ψ ) generated by ψ ∈ 10 C02 (R) (we refer to [5] for more details). propChen
Proposition 2.4 For a C 2 function ψ : R → R, compactly supported on the interval [a, b], we have: supp(η ψ ), supp(q ψ ) ⊂ {(ρ, m) = (ρ, ρu) : u + ρθ ≥ a, u − ρθ ≤ b} :
(2.26)
Furthermore, there exists a constant Cψ such that, for any ρ ≥ 0 and u ∈ R, we have: 9
3.2
• For γ ∈ (1, 3], |η ψ (ρ, m)| + |q ψ (ρ, m)| ≤ Cψ ρ.
(2.27)
3.3
|η ψ (ρ, m)| ≤ Cψ ρ, |q ψ (ρ, m)| ≤ Cψ (ρ + ρθ+1 ).
(2.28)
3.4
(2.29)
3.5
(2.30)
3.6
• For γ ∈ (3, +∞),
• If η ψ is considered as a function of (ρ, m), m = ρu then ψ ψ |ηm (ρ, m)| + |ρηmm (ρ, m)| ≤ Cψ , ψ and, if ηm is considered as a function of (ρ, u), then ψ ψ |ηm (m, u)| + |ρ1−θ ηmρ (ρ, ρu)| ≤ Cψ .
1.3
35
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We now would like to express the kinetic formulation of (1.5) introduced in ([26]). Theorem 2.4 Let (ρ, ρv) ∈ L∞ (R+ , L1 (R)) have finite energy and ρ ≥ 0, then it is an 1.3 entropy solution of (1.5) if and only if there exists a non-positive bounded measure m on R+ × R2 such that the function χ(ρ, ξ − u) satisfies: ∂t χ + ∂x [(θξ + (1 − θ)u)χ] = ∂ξξ m(t, x, ξ).
(2.31)
cinetique theo
3
Proof of theorem 1.1
section3
Sol
We would like to start with recalling an important result due to Solonnikov (see [34]). Let ρ0 the initial density such that: 0 < m0 ≤ ρ0 ≤ M0 < +∞.
(3.32)
initiald
(3.33)
visco
When the viscosity coefficient µ(ρ) satisfies: µ(ρ) ≥ c > 0 for allρ ≥ 0, we have the existence of strong solution for small time. More exactly, we have: initiald
visco
Proposition 3.5 Let (ρ0 , v0 ) satisfy (3.32) and assume that µ satisfies (1.1 3.33), then there exists T0 > 0 depending on m0 , M0 , kρ0 − ρ¯kH 1 and kv0 kH 1 such that (1.3) has a unique solution (ρ, v) on (0, T0 ) satisfying: ρ − ρ¯ ∈ L∞ (H 1 (R), ∂t ρ ∈ L2 ((0, T1 ) × R), v ∈ L2 (0, T1 , H 2 (R)), ∂t v ∈ L2 ((0, T1 ) × R) for all T1 < T0 . Remark 3 The main point in this theorem is that the time of existence T depends only 1.1 0 of the norms of ρ0 which gives us a low bounds on T0 of the system (1.3). 10
In view of this proposition, we see that if we introduce a truncated viscosity coefficient µn (ρ): 1 µn (ρ) = max(ρ, ), n then there exists approximated solutions (ρ , v ) defined for small time (0, T0 ) of the 1.1 theo n n system (1.3). In order to prove theorem 1.1 , we only have to show that (ρn , vn ) satisfies the following bounds uniformly with respect to n and T large: 0 < m0 ≤ ρn ≤ M0 < +∞, ∀t ∈ [0, T ], 1 ρn − ρ¯ ∈ L∞ T (H (R)),
vn ∈
(3.34)
1 L∞ T (H (R)).
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36
We are going to follow the method of Lions et al in [25], indeed the main point is to prove that we can extend the notion of Riemann invariant or more precisely1.1 the kinetic cinetique 1.1 formulation of proposition 2.4 to the system (1.3). We recall that system (1.3) has the following form: ∂ ρn + ∂x (ρn vn ) − ∂xx ρn = 0, ∂t (3.35) ∂ (ρn vn ) + ∂x (ρn vn vn ) − ∂x (∂x ρn vn ) − ∂x (ρn ∂x vn ) + ∂x (a(ρn )γ = 0, ∂t and we have finally: ∂ ρn + ∂x (ρn vn ) − ∂xx ρn = 0, ∂t ∂ (ρn vn ) + ∂x (ρn vn vn ) − ∂x ∂x (ρn vn ) + ∂x (a(ρn )γ = 0, ∂t
(3.36)
1.1a
1.1b
36
Following [25]entropie and settingflux mn = ρn vn we have for any pair of entropy flux (η(ρ, u), H(ρ, u)) defined by (2.23) and (2.24) where η is a convex function of (ρn , mn ). We write η = η¯(ρn , mn ): ∂t η + ∂x H = ¯ ηρ ∂xx ρn + ¯ ηm ∂xx mn , = ∂xx η − (¯ ηρρ (∂x ρn )2 + 2¯ ηρm (∂x ρn )(∂x mn ) + η¯mm (∂x mn )2 ). Here we define µn such that: µn = η¯ρρ (∂x ρn )2 + 2¯ ηρm (∂x ρn )(∂x mn ) + η¯mm (∂x mn )2 proputile
By proposition 2.2, we can check that µn ≥ 0. We obtain then that: ∂t η(ρn , vn ) + ∂x H((ρn , vn ) − ¯ ηρ ∂xx ρn ≤ 0 in R × (0, +∞). cinetique
By applying the same method than for proving the theorem 2.4, we obtain the following kinetic formulation: ∂t χ + ∂x ([θξ + (1 − θ)vn ]χ) − ∂xx χ = ∂ξξ m ¯ n on R2 × (0, +∞),
11
(3.37)
riemann
where m ¯ n is a nonpositive bounded measure riemann on R2 × (0, +∞). Finally we recover the classical maximum principle by multiplying (3.37) by the convex functions g(ξ) = (ξ − ξ0 )+ and g(ξ) = (ξ − ξ0 )− and integrating over R2 × (0, +∞). Indeed as we have that: −C ≤ min(v0 − ρθ0 ) ≤ max(v0 + ρθ0 ) ≤ C, x
x
and that: suppξ = [v − ρθ , v + ρ[θ]. For ξ0 large enough, we can show that: suppξ0 ∩ suppχ = ∅. We have obtain then that: −C ≤ min(v0 − ρθ0 ) ≤ vn − ρθn ≤ vn + ρθn ≤ max(v0 + ρθ0 ) ≤ C.
hal-00635983, version 1 - 26 Oct 2011
x
x
In particular we obtained that ρn and vn are uniformly bounded in L∞ (0, Tn , L∞ (R)) or: |ρn (t, x)| + |vn (t, x)|) ≤ C0 ,
sup
(3.38)
imp2
x∈R,t∈(0,Tn )
theo1
4 4.1 section4
Proof of theorem 1.2 1.1
Uniform estimates for the solutions of (1.3) 1.1
First we assume thattheo (ρ , v ) is the global solutions of Korteweg’s equations (1.3) constructed in theorem 1.1 and satisfying: ρ (t, x) ≥ c (t), for some c (t) > 0,
(4.39)
2.1
(4.40)
2.2
and lim (ρ , v )(x, t) = (ρ± , u± ).
x→±∞
Here we are working around a non constant state (¯ ρ, v¯) with: lim (¯ ρ, v¯)(x, t) = (ρ± , u± ).
x→±∞
theo
1.8
It1.9 is a simple1.10 extension of theorem 1.1. Our goal is now to check the properties (10 1.11), theo2 (1.12) and (1.13) in order to use the theorem 1.3 of Chen and Perepelista (see [5]) in theo1 order to prove the theorem 1.2. For simplicity, throughout this section, we denote (ρ, v) = (ρ , v ) and C > 0 denote the constant independent of . 1.1 We start with recalling the inequality energy for system ( 1.3), indeed by the introduction Hprepa of the effective velocity we obtain new entropies (see [13]). Lemma 1 Suppose that E1 [ρ0 , u0 ] ≤ E0 < +∞ for some E0 > 0 independent of . It holds that: Z tZ sup E1 [ρ, u](τ ) + ρu2x dxdτ ≤ C(t), (4.41) 0≤τ ≤t
0
12
R
2.3a
and: Z tZ
ρvx2 dxdτ +
sup E2 [ρ, v](τ ) + 0≤τ ≤t
lemma1
0
Z tZ 0
R
ργ−2 ρ2x dxdτ ≤ C(t),
(4.42)
2.3
R
where C(t) depends on E0 , t, ρ¯, and u ¯ but not on . 1.1
Proof: It suffices to writes the energy inequalities for system (1.3) and from system 1.3 (1.5). More exactly we have: Z Z d (η(ρ, m) − η(¯ ρ, ρ¯u ¯)dx + ρu2x dx = q(ρ− , m− ) − q(ρ+ , m+ ), dt R R with the entropy pair:
hal-00635983, version 1 - 26 Oct 2011
η(ρ, m) = with e(ρ) =
a γ γ−1 ρ .
m2 m3 0 + e(ρ), q(ρ, m) = 2 + me (ρ), 2ρ 2ρ
Since we have: (ρ, ρ¯) ≥ ρ(ρθ − ρ¯θ )2 , θ =
γ−1 , 2
we can classically bootstrap on the left hand-side the term q(ρ− , m− ) − q(ρ+ , m+ ). Remark 4 Since vacuum could occur in our solution, the inequality Z tZ ρu2x dxdτ ≤ C(t), 0
R
2.3
10
lemme2
in (4.42) is much weaker than the corresponding one in [5]. That is why lemma 2 will be more tricky to obtain. The following higher order integrability estimate is crucial in compactness argument. lemma1
lemme2
Lemma 2 If the conditions of lemma 1 hold, then for any −∞ < a < b < +∞ and all t > 0, it holds that: Z tZ b ργ+1 dxdτ ≤ C(t, a, b), (4.43) 0
2.21
a
where C(t) > 0 depends on E0 , a, b, γ, t, ρ¯, u ¯ but not on . Remark 5 The proof follows the same ideas than in the case of compressible NavierStokes equations when we wish to obtain a gain of integrability on the density. We refer fL2 Hu2 to [24] for more details. The proof is also inspired from Huang et al in [19]. Proof. Choose ω ∈ C0∞ (R) such that: 0 ≤ ω(x) ≤ 1, ω(x) = 1 for x ∈ [a, b], and suppω = (a − 1, b + 1). 1.1
By the momentum equation of (1.3) and by localizing, we have (P (ρ)ω)x = −(ρuvω)x + (P (ρ) + ρuv)ωx − (ρv)t ω + (ρvx ω)x − ρvx ωx . 13
(4.44)
2.22
2.22
Integrating (4.44) with respect to spatial variable over (−∞, x), we obtain: Z x Z x [(ρuv + P (ρ))ωx − ρvx ωx . (4.45) ρv ωdy)t + P (ρ)ω = −ρuvω + (ρvx ω)x − (
2.23
−∞
−∞
2.23
Multiplying (4.45) by ρω, we have 2
2
2
2
x
Z
2
ρP (ρ)ω = − ρ uvω + ρ vx ω − (ρω ρv ωdy)t −∞ Z x Z x [(ρuv + P (ρ))ωx − ρvx ωx ]dx, ρu ωdy) + ρω − (ρu)x ω( −∞ −∞ Z x Z x 2 2 ρv ωdy)x ρv ωdy)t − (ρuω =ρ vx ω − (ρω −∞ −∞ Z x Z x [(ρuv + P (ρ))ωx − ρvx ωx ]dx, ρv ωdy + ρω + ρuωx
2.24
(4.47)
2.25
(4.48)
2.26
(4.49)
2.27
−∞
−∞
hal-00635983, version 1 - 26 Oct 2011
(4.46)
2.24
We now integrate (4.46) over (0, t) × R and we get: Z tZ aρ 0
Z tZ
γ+1 2
0
+
Z
R
Z
−∞
Z
Z tZ
Z
R
ρv ωdy dxdτ
−∞
R
x
[(ρuv + P (ρ))ωx − ρvx ωx ]dx dxdτ.
ρω 0
ρv ωdy)dx
x
ρuωx 0
x
−∞
R
ρ0 v0 ωdy)dx + +
Z (ρω
Z tZ
x
(ρ0 ω R
2
ρ vx ω −
ω dxdτ =
R
Z
2
−∞
Let A = {x : ρ(t, x) ≥ ρ}, where ρ = 2 max(ρ+, ρ−), 2.3
then we have the following estimates by (4.42): |A| ≤
C(t) e∗ (2ρ, ρ¯)
= d(t).
2.26
By (4.48), for any (t, x) there exists a point x0 = x0 (t, x) such that |x − x0 | ≤ d(t) and ρ(t, x0 ) = ρ. Here we choose β = γ+1 2 > 0, suppx∈supp(ω) ρβ (t, x) ≤ ρβ + suppx∈supp(ω)∩A ρβ (t, x), ≤ 2ρβ + suppx∈supp(ω)∩A |ρβ (t, x) − ρβ (t, x0 )|, Z x0 +d(t) β ≤ 2ρ + suppx∈supp(ω)∩A |β||ρβ−1 (t, x)ρx |dx, x0 −d(t)
β
Z
b+1+2d(t) 2β−1
≤ 2ρ +
|β|ρ a−1−2d(t)
Z
dx + R
b+1+2d(t)
≤ C(t) +
Z
ργ dx,
a−1−2d(t)
≤ C(t). 14
2 ρ−1 ρ2x dx,
(4.50)
2.28
2.28
Using (4.50), 2.25 Young inequalities and H¨older’s inequalities, the first term on the right hand side of (4.47) is treated as follows: Z tZ ρ2 vx ω 2 dxdτ 0 R Z tZ Z tZ 1 1 ≤ ρ3 ω 4 dxdτ + ρv 2 dxdτ, 2 0 R 2 0 R x Z tZ (4.51) ρ3 ω 2 dxdτ, ≤ C(t) + 0 R Z tZ ≤ C(t) + C(t) ρ4−β ω 2 dxdτ, 0 R Z tZ ργ+1 ω 2 dxdτ, ≤ C(t) + δ 0
R
lemma1
5 3.
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2.29
Here we have used the fact that γ > By lemma 1 and the H¨older inequality, we obtain Z Z x |ρv|dy, ρvωdy ≤ supp(ω) −∞ Z Z (4.52) 1 1 2 2 2 ≤( ρdy) ( ρv dy) ≤ C(t). supp(ω)
2.30
supp(ω)
Then:
Z
Z
x
ρvωdy dx +
ρω R
Z
Z
−∞
ρ0 v0 ωdy dx
−∞
R
+
x
ρ0 ω Z tZ
Z
x
ρuωx 0
(4.53)
2.32
(4.54)
2.33
(4.55)
2.34
ρvωdy dxdτ ≤ C(t).
−∞
R
Similarly, we have:
Z tZ
Z
x
(ρuv + P (ρ))ωx dy dxdτ ≤ C(t),
ρω 0
and
−∞
R
Z tZ
Z
x
ρvx ωx dy dxdτ
ρω 0
≤
Z
ρ|vx | |ωx |dy dxdτ , R R Z Z Z 2 ρωdx ρvx dy + ρωx2 dy dτ , ρω
0
≤
−∞
R
Z tZ t
Z 0
R
R
R
≤ C(t). 2.29
2.32
2.34
2.25
Substituting (4.51), (4.53)-(4.55) into (4.47) and noticing the smallness of δ, we proved lemme2 lemma 2. lemma1
Lemma 3 Suppose that (ρ0 (x), v0 (x) satisfy the conditions in the lemmas 1. Furthermore there exists M0 > 0 independent of , such that Z ρ0 (x)|v0 (x) − v¯(x)|dx ≤ M0 < +∞, (4.56) R
15
2.35
then for any compact set K ⊂ R, it holds that: Z tZ (ργ+θ + ρ|v|3 )dxdτ ≤ C(t, K), 0
(4.57)
2.36
K
where C(t, K) is independent of . 2.36
Remark 6 In order to prove the inequality (4.57), we will use the same ingredients than 35 in [26] where this inequality was obtained for the first time. pair35
hal-00635983, version 1 - 26 Oct 2011
ψ Proof. We are now working with the function ψ of proposition 2.3. If we consider ηm as a function depending of (ρ, v), we have for all ρ ≥ 0 and v ∈ R: ( ψ |ηmv (ρ, v)| ≤ C, (4.58) ψ |ηmρ (ρ, v)| ≤ Cρθ−1 .
2.38
For this weak entropy pair (η ψ , H ψ ), we observe that: ψ
η (ρ, 0) =
ηρψ (ρ, 0)
θ = 0, H (ρ, 0) = ρ3θ+1 2 ψ
and: ψ ηm (ρ, 0)
Z
θ
= βρ with β =
Z
|s|3 [1 − s2 ]λ+ ,
R
|s|[1 − s2 ]λ+ ds.
R
By Taylor formula, we have: η ∗ (ρ, m) = βρθ m + r(ρ, m),
(4.59)
2.39
r(ρ, m) ≤ Cρv 2 ,
(4.60)
2.40
(4.61)
2.41
(4.62)
2.42
(4.63)
2.43
with: b such that, for some constant C > 0. Now we introduce a new entropy pair (b η , H) b m) = H ψ (ρ, m − ρv − ) + v − η ψ (ρ, m − ρv − ), ηb(ρ, m) = η ψ (ρ, m − ρv − ), H(ρ, with m = ρv which satisfies: ( ηb(ρ, m) = βρθ+1 (v − v − ) + r(ρ, ρ(v − v − )), r(ρ, ρ(v − v − )) ≤ Cρ(v − v − )2 .
1.1
1.1
Integrating (1.3)1 × ηbρ + (1.3)2 × ηbm over (0, t) × (−∞, x), we have: Z x Z t ηb(ρ, m) − ηb(ρ0 , m0 ) dy + q ∗ (ρ, ρ(v − v − )) + v − η ∗ (ρ, ρ(v − v − ))dτ −∞
0
∗
−
Z
= tq (ρ , 0) + 0
t
Z tZ ηbm ρvx dτ −
0
x
−∞
(b ηmu ρvx2
+ ηbmρ ρρx vx )dydτ.
2.38
By using (4.58), we obtain: Z tZ x Z tZ ηbmu ρvx2 dydτ ≤ C ρvx2 dy dτ ≤ C(t), 0
−∞
0
16
R
Z tZ
x
ηbmρ ρρx vx dydτ ≤ C
−∞
0
Z tZ ≤ C 0
2.43
0
ρvx2 dy dτ
ρθ−1 ρ|ρx vx |dy dτ ≤ C(t),
R
Z tZ + C 0
R
2.44
Z tZ
(4.64) ργ−2 ρ2x dy dτ
≤ C(t).
R
2.42
2.37
Substituting (4.63) and (4.64) into (4.62), then integrating over K and using (2.25), we obtain: Z tZ ρθ+γ + ρ|v − v − |3 dxdτ 0 K Z tZ Z tZ ∗ − ρ|v||vx |dxdτ |η (ρ, ρ(v − v )|dxdτ + C ≤ C(t) + C (4.65) 0 K 0 K Z tZ Z Z x 1+θ c ρ |vx |dxdτ + 2 sup ( eta(ρ(y, τ ), (ρv)(y, τ ))dy)dx . + C 0
τ ∈[0,t]
K
2.44
2.45
−∞
K
hal-00635983, version 1 - 26 Oct 2011
lemma1
Applying lemma 1, we have: Z tZ
|η ∗ (ρ, ρ(v − v − )|dxdτ ≤ C(t).
(4.66)
2.46
(4.67)
2.47
(4.68)
2.48
Now we are going to deal with the last term on the right hand side of (4.65). (1.3) implies that: (ρv − ρv − )t + (ρv 2 + P (ρ) − ρuu− )x = (ρvx )x . (4.69)
2.49
0
K
2.28
By H¨older’s inequality and (4.50), we get: Z tZ
1+θ
ρ 0
Z tZ |vx |dxdτ ≤ C
K
0
ρvx2 dxdτ
Z tZ 0
Z tZ
K
ρθ dxdτ,
≤ C(t) + C(t) 0
ρ1+2θ dxdτ,
+ C
K
K
≤ C(t). We have now: Z tZ
Z tZ Z tZ 1 1 ρ|v||vx |dxdτ ≤ ρvx2 dxdτ + ρv 2 dxdτ, 2 2 K 0 K 0 K ≤ C(t).
0
2.45
1.1
2.49
Integrating (4.69) over [0, t] × (−∞, x) for x ∈ K, we get: Z
x
−
Z
x
ρ(v − ρv )dy = −∞
Z
−
ρ0 (v0 − ρv )dy − −∞
t
(ρv 2 + P (ρ) − ρuu− − P (ρ− ))
0
Z +
ρvx dτ. 0
17
(4.70)
t
2.50
Furthermore: Z x ηb((ρ(y, τ ), (ρv)(y, τ ))dy −∞ Z x Z x βρθ+1 (v − v¯))dy ≤ (b η (ρρv) − βρθ+1 (v − v¯))dy + | −∞ Z−∞ Z x x (r(ρρ(v − v¯))dy + β(ρθ − (ρ− )θ )ρ(v − v¯))dy ≤ −∞ −∞ Z x ρ(v − v¯))dy , + β(ρ− )θ −∞ Z x ρ(v − v¯))dy . ≤ C(t) + β(ρ− )θ
(4.71)
2.51
−∞
2.35
lemma1 lemme2 2.50
2.51
hal-00635983, version 1 - 26 Oct 2011
By using (4.56), lemma 1 and 2, (4.70) and (4.71) we conclude the proof of the lemma.
4.2
−1 Hloc (R2+ ) Compactness
In this section we are going to take profit of the uniform estimates obtained in the previous −1 (R2+ )-compactness of section in order to prove the following lemma, which gives the Hloc the Korteweg solution sequence (ρ , v ) on a entropy- entropy flux pair. Lemma 4 Let ψ ∈ C02 (R), η ψ , H ψ ) be a weak entropy 1.1 pair generated by ψ. Then for the solutions (ρ , v ) with m = ρ v of Korteweg system (1.3, the following sequence: −1 (R2+ ) η ψ (ρ , m )t + q ψ (ρ , m )x are compact in Hloc
(4.72)
3.1
lemme4 1.1
Proof: Now we are going to prove the lemma. A direct computation on (1.3)1 × 1.1 ψ ηρψ (ρ , m ) + (1.3)2 × ηm (ρ , m ) gives: ψ ηρψ (ρ , m )t + Hρψ (ρ , m )x = ηρψ (ρ , m )(ρ )vx − ηmu (ρ , m )(ρ )(vx )2 (4.73) ψ − ηmu (ρ , m )(ρ )vx ρx .
3.7
propChen 3.6
Let K ⊂ R be compact, using proposition 2.4 (2.30) and H¨older inequality, we get: Z tZ ψ ψ |ηmu (ρ , m )(ρ )|(vx )2 + |ηmu (ρ , m )(ρ )vx ρx |dxdt 0 K Z tZ Z tZ (4.74) (ρ )|(vx )2 dxdτ + C (ρ )γ−2 (ρx )2 dxdτ ≤ C 0
K
0
3.8
K
≤ C(t). This shows that: ψ ψ −ηmu (ρ , m )ρ (vx )2 − ηmu (ρ , m )ρ vx ρx are bounded in L1 ([0, T ] × K), −1,p1 and thus it is compact in Wloc (R2+ ), for 1 < p1 < 2. Moreover we observe that ψ |ηmu (ρ , ρ v )| ≤ Cψ ,
18
(4.75)
3.9
, then we obtain: Z tZ
4
ψ (ηm (ρ , m )ρ vx ) 3 dxdt 0 K Z tZ 4 4 4 3 |ρ | 3 |vx | 3 dxdt ≤ 0 K Z tZ Z tZ 4 4 2 ρ |vx | dxdt + C 3 (ρ )2 dxdt ≤ C 3 0 K 0 K Z tZ 1 4 (ρ )γ+1 dxdt →→0 0. ≤ C(t, K) 3 + C 3 0
3.10
(4.76)
3.10
(4.77)
3.11
(4.78)
3.12
K
3.9
Using (4.76) and (4.75), we obtain −1,p1 (R2+ ) for some 1 < p1 < 2. ηρψ (ρ , m )t + Hρψ (ρ , m )x are compact in Wloc
3.3
3.4
lemma1 lemme2 2.36
hal-00635983, version 1 - 26 Oct 2011
Furthermore by (2.27)-(2.28), lemma 1-2 and (4.57), we have: 3 ηρψ (ρ , m )t + Hρψ (ρ , m )x are uniformly bounded in Lploc (R2+ ) for p3 > 2.
where p3 = γ + 1 > 2 when γ ∈ (1, 3], and p = lemme4 3 we conclude the proof of the lemma 4.
γ+θ 1+θ
> 2 when γ > 3. By interpolation
theo1
5
Proof of theorem 1.2 lemma1
theo2
From lemmas 1, we have verified the theo2 conditions (i)-(iii) of theorem 1.3 for the sequence of solutions (ρ , m ). Using theorem 1.3, there exists a subsequence (ρ , m ) and a pair of measurable functions (ρ, m) such that (ρ , m ) → (ρ, m), a.e → 0.
(5.79)
It is easy 1.3 to check that (ρ, m) is a finite-energy entropy solution (ρ, m) to the Cauchy 5 problem (1.5) with initial data (ρtheo1 0 , ρ0 u0 ) for the isentropic Euler equations with γ > 3 . It achieves the proof of theorem 1.2.
References fA
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[4] G. Q. Chen, The theory of compensated compactness and the system of isentropic gas dynamics, Lecture notes, Preprint MSRI-00527-91, Berkeley, October 1990. 19
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[28] L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98, no. 2, 123-142 (1987).
Mu
[29] F. Murat, Compacit´e par compensation, Ann. Sci. Norm. Sup. Pisa. 5, 1978, p 489-507.
Per
[30] B. Perthame, Higher moments lemma: application to Vlasov-Poisson and FokkerPlanck equations. Math. Meth. Appl. Sc. 13, 441-452 (1990).
Rohdehdr
[31] C. Rohde, Approximation of Solutions of Conservation Laws by Non- Local Regularization and Discretization, Habilitation Thesis, University of Freiburg (2004).
fR
[32] J.S. Rowlinson, Translation of J.D van der Waals. The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density. J.Statist. Phys., 20(2): 197-244, 1979.
Serre
[33] D. Serre, Systems of conservation laws, I- Hyperbolicity, Entropies, Shock Waves, Cambridge University Press (1999).
Sol
[34] V. A Solonnikov. The solvability of the initial boundary-value problem for the equations of motion of a viscous compressible fluid. Zap Naucn. Sem. Leningrad. Odel. Mat. Inst. Steklov. (LOMI), 56: 128-142, 197, 1976. Investigations on linear operators and theory of functions, VI.
Ta
[35] L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics , Herriot-Watt Symposium, Vol 4, R. J. Knops, ed, Research Notes in Mathematics No 39, Pitman, Boston-London, 1979, p 136-212. 21
[36] L. Tartar, The compensated compactness method applied to systems of conservation laws, Systems of Nonlinear Partial Differential Equations, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 111, J. Bali, ed, Reidel, Dordrecht-Boston, 1983, p 263-285.
fTN
[37] C. Truedelland W. Noll. The nonlinear field theories of mechanics. Springer-Verlag, Berlin, second edition, 1992.
VW
[38] J.F Van der Waals. Thermodynamische Theorie der Kapillarit¨at unter Voraussetzung stetiger Dichte¨ anderung, Phys. Chem. 13, 657-725 (1894).
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Ta1
22
†‡
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Abstract In the first part of this paper, we prove the existence of global strong solution for Korteweg system in one dimension. In the second part, motivated by the processes of vanishing capillarity-viscosity limit in order to select the physically relevant solutions for a hyperbolic system, we show that the global strong solution of the Korteweg system converges in the case of a γ law for the pressure (P (ρ) = aργ , γ > 1) to entropic solution of the compressible Euler equations. In particular it justifies that the Korteweg system is suitable for selecting the physical solutions in the case where the Euler system is strictly hyperbolic. The problem remains open for a Van der Waals pressure because in this case the system is not strictly hyperbolic and in Lax,G particular the classical theory of Lax and Glimm (see [21, 11]) can not be used.
1
Introduction
We are concerned with compressible fluids endowed with internal capillarity. The model we consider originates from the XIXth century work by Van der Waals and Korteweg VW,fK [38, 22] and was actuallyfDS,fJL,fTN derived in its modern form in the 1980s using the second gradient theory, see for instance [9, 20, 37]. The first investigations begin with the Young-Laplace theory which claims that the phases are separated by a hypersurface and that the jump in the pressure across the hypersurface is proportional to the curvature of the hypersurface. The main difficulty consists in describing the location and the movement of the interfaces. Another major problem is to understand whether the interface behaves as a discontinuity in the state space (sharp interface) or whether the phase boundary corresponds to a more regular transition (diffuse interface, DI). The diffuse interface models have the advantage to consider only one set of equations in a single spatial domain (the density takes into account the different phases) which considerably simplifies the mathematical and numerical study (indeed in the case of sharp interfaces, we have to treat a problem with free boundary). ∗
Universit´e Paris-Est Cr´eteil, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees (UMR 8050), 61 Avenue du G´en´eral de Gaulle, 94 010 Cr´eteil Cedex (France). E-mail: [email protected] † Basque Center of Applied Mathematics, Bizkaia Technology Park, Building 500, E-48160, Derio (Spain) ‡ Ceremade UMR CNRS 7534 Universit´e de Paris IX- Dauphine, Place du Marchal DeLattre De Tassigny 75775 PARIS CEDEX 16 , [email protected]
1
Let us consider a fluid of density ρ ≥ 0, velocity field u ∈ R, we are now interested in the following compressible capillary fluid model, which can be derived from a fDS Cahn-Hilliard like free energy (see the pioneering work by J.E. Dunn and J. Serrin in [9] and also in fA,fC,fGP,HM [1, 3, 12, 17]). The conservation of mass and of momentum write: ∂ ρ + ∂x (ρ u ) = 0, ∂t (1.1) ∂ (ρ u ) + ∂x (ρ (u )2 ) − ∂x (ρ ∂x u ) + ∂x (a(ρ )γ ) = 2 ∂x K, ∂t
3systeme
where the Korteweg tensor reads as following:
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1 0 divK = ∂x ρ κ(ρ )∂xx ρ + (κ(ρ ) + ρ κ (ρ ))|∂x ρ |2 − ∂x κ(ρ )(∂x ρ )2 . 2
(1.2)
κ is the coefficient of capillarity and is a regular function of the form κ(ρ) = 2 ρα with 2 α ∈ R. In the sequel we shall assume that κ(ρ) = ρ . The term ∂x K allows to describe the variation of density at the interfaces between two phases, generally a mixture liquidvapor. P = aργ with γ ≥ 1 is a general γ law pressure term. corresponds to the controlling parameter on the amplitude of the viscosity and of the capillarity.Hprepa When we 3systeme set v = u + ∂x (ln ρ ), we can write (1.1) on the following form (we refer to [13] for the computations): ∂ ρ + ∂x (ρ v ) − ∂xx ρ = 0, ∂t (1.3) ∂ (ρ v ) + ∂x (ρ (u )(v )) − ∂x (ρ ∂x v ) + ∂x (a(ρ )γ ) = 0, ∂t
divK
1.1
1.1
We now consider the Cauchy problem of (1.3) when the fluid is away from vacuum. 1.1 Namely, we shall study (1.3) with the following initial data: ρ (0, x) = ρ0 (x) > 0, u (0, x) = u0 (x),
(1.4)
1.2
such that: lim
x→+,−∞
(ρ0 (x), u0 (x)) = (ρ+,− , u+,− ), with ρ+,− > 0. 1.1
We would like to study in the sequel the limit process of system (1.3) when goes to 0 and to prove in particular that we obtain entropic solutions of the Euler system: ∂ ρ + ∂x (ρv) = 0, ∂t (1.5) ∂ (ρv) + ∂x (ρv 2 ) + ∂x (aργ ) = 0, ∂t Let us now explain the interest of the capillary solutions for the hyperbolic systems of conservation laws.
1.1
Viscosity capillarity processes of selection for the Euler system
In addition of modeling a liquid-vapour mixture, the Korteweg also shows purely theoretical interests consisting in the selection of the physically relevant solutions of the 2
1.3
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Euler model (in particular when the system is not strictly hyperbolic). The typical case corresponds to a Van der Waals pressure: indeed in this case the system is not strictly hyperbolic in the elliptic region (which corresponds to the region where the phase change occurs). In the adiabatic pressure framework (P (ρ) = ργ with γ > 1), the system is strictly hyperbolic and the theory is classical. More precisely we are able to solve the Riemann problem when the initial Heaviside data is small in the BV space. Indeed we are in the context of the well known Lax result as the system is also genuinely nonlinear (we refer Lax to [21]). It means we have existence of global C 1 -piecewise solutions which are unique in the class of the entropic solutions. This result as been extent by Glimm in the context of small initial data in the BV-space by using a numerical scheme and approximating the initial BV data by a C 1 -piecewise function (which implies to locally solve the Riemann problem via the Lax result). For BB1 the uniqueness of the solution we refer to the work of Bianchini and Bressan ([2]) who use a viscosity method. In the setting of the Van der Waals pressure, the existence of global solutions and the nature of physical relevant solutions remain completely open. Indeed the system is not strictly hyperbolic anymore. If we rewrite the compressible Euler system in Lagrangian coordinates by using the specific volume τ = 1/ρ in ( 1b , ∞) and the velocity u, the system satisfies in (0, +∞) × R the equations: ∂t τ − ∂x u = 0, (1.6) ∂t u − ∂x (Pe(τ )) = 0,
euler
with the function Pe : ( 1b , ∞) → (0, ∞) given by: 1 Pe(τ ) = P ( ), τ The two eigenvalues of the system are: q λ1 (τ, v) = − −Pe0 (τ ),
1 τ ∈ ( , ∞). b
q λ2 (τ, v) = − −Pe0 (τ ).
The corresponding eigenvectors r1 , r2 are: ! 1 q w1 (τ, v) = , w2 (τ, v) = −Pe0 (τ )
q 1 − −Pe0 (τ )
(1.7)
! (1.8)
Furthermore by calculus we obtain: 00 00 −Pe (τ ) Pe (τ ) ∇λ1 (τ, v) · w1 (τ, v) = q , ∇λ2 (τ, v) · w2 (τ, v) = q 2 −Pe0 (τ ) 2 −Pe0 (τ )
(1.9)
We now recall the definition of a standard conservation law in the sense of Lax (it means entropy solutions): • The system is strictly entropic if the eigenvalues are distinct and real. 3
vp
• The characteristics fields are genuinely nonlinear if we have for all (τ, v), ∇λ1 (τ, v) · w1 (τ, v) 6= 0 and ∇λ2 (τ, v) · w2 (τ, v) 6= 0, Serre
for more details we refer to [33]. The definition of genuine nonlinearity is some kind of extension of the notion of convexity to vector-valued functions (in particular when we consider the specific case of the traveling waves). The previous assumptions aim at Evans Serre ensuring the existence and the uniqueness of the Riemann problem ( see [10] and [33]). When P is a Van der Waals pressure, we observe that the first conservation law Serre,Evans ([33, 10]) is far from being a standard hyperbolic system, indeed: • It is not hyperbolic (but elliptic) in ( α11 , α12 ) × R,
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• the characteristic fields are not genuinely nonlinear in the hyperbolic part of the state space. Here the classical Lax-Glimm theory cannot be applied. In particular there doesn’t exist any entropy-flux pair, which suggests that the entropy framework is not adapted for selecting the physically relevant solutions. In order to deal with this problem, Van der Waals and Korteweg began by considering the stationary problem with null velocity, and Rohdehdr solving ∇P (ρ) = 0. For more details we refer to [31]. It consists in minimizing in the following admissible set Z 1 1 A0 = {ρ ∈ L (Ω)/W (ρ) ∈ L (Ω), ρ(x)dx = m}, Ω
the following functionnal Z F [ρ] =
W (ρ(x))dx. Ω
Unfortunately this minimization problem has an infinity of solutions, and many of them are physically irrelevant. In order to overcome this difficulty, Van der Waals in the XIXth century was the first to regularize the previous functional by adding a quadratic term in the density gradient. More precisely he considered the following functional: Z 2 Flocal = W (ρ (x)) + γ |∇ρ |2 dx, 2 Ω with: Alocal = H 1 (Ω) ∩ A0 . This variational problem has a unique solution and its limit (as goes to zero) converge to a physical solution of the equilibrium problem for the Euler system with Van der Waals REF pressure, that was proved by Modica in [28] with the use of gamma-convergence. By the Euler-Lagrange principle, the minimization of the Van der Waals functional consists in solving the following stationary problem: ∇P (ρ ) = γ2 ρ ∇∆ρ , where the right-hand side can be expressed as the divergence of the capillarity tensor. Heuristically, we also hope that the process of vanishing capillarity-viscosity limit selects the physical relevant solutions as it does for the stationary system. This problem actually remains open. 4
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1.2
Existence of global entropic solutions for Euler system
Before presenting the results of this paper let us recall the results on this topic in these last decades. We shall focus on the case of a γ pressure law P (ρ) = aργ with γ > 1 and a positive. Let us mention that these cases are the only ones well-known (essentially because the system is strictly hyperbolic in this case and that we can exhibit many entropy-flux pairs). Here the Lax-Glimm theory can be applied, however at the end of the 70’s, one was interested in relaxing the conditions on the initial data by only assuming ρ0 and u0 in L∞ . In the beginning of the 80’s Di Perna initiated this program, consisting in obtaining global entropic solutions for L∞ initial data. Di1, Di2 1.3 Indeed in [7, 8], Di Perna prove the existence of global weak entropy solution of (1.5) 2 3 for γ = 1 + 2d+1 and γ = 2k + 2k + 1 (with k ≥ Ta 1), d ≥ 2 by using the so-called ”compensated compactness” introduced by Tartar in 35 [35]. This result was extended36by Chen Chen in [4] in the case γ ∈ (1, 35 ] and by Lions et al in [26] in the case γ ∈ [3, ∞). In [25], Lions et al generalize this result to the general case γ ∈ (1, 3), and finally the case γ = 1 Hu1 is treated by [18]. We would like to mention that these results are obtained through a vanishing artificial viscosity on both density and velocity. The problem of vanishing physical viscosity limit of compressible Navier-Stokes equations to compressible10Euler equations was until recently an open problem. However Chen and Perepelista in [5] proved that the solutions of the compressible Navier-Stokes system with constant viscosity coefficients converge to aHu2 entropic solution of the Euler system with finite energy. This result was extended in [19] to the case of viscosity coefficients depending on the density. 10 Hu2 Inspired by [5] and [19], we would like to show that the solution of the Korteweg system 1.1 (1.3) converges to a entropic solution of the Euler system with finite energy when the pressure is a γ law. To do this, we will prove for the first time up our knowledge the existence of global strong solution for the Korteweg system in one dimension in the case of Saint-Venant viscosity coefficients. By contrast, the problem of global strong solutions for compressible Navier-Stokes equations remains open (indeed one of the main difficulties consists in controlling the vacuum). This result justifies that the Korteweg system allows us to select the relevant physical solutions of the compressible Euler system at least when the pressure is adiabatic (P (ρ) = aργ with γ > 1). The problem remains open in the case of a Van der Waals pressure.
1.3
Results
Let us now describe our main result. In the1.1 first theorem we prove the existence of global strong solution for the Korteweg system (1.3). Theorem 1.1 Let ρ¯ > 0. Assume that the initial data ρ0 and u0 satisfy: 0 < m0 ≤ ρ0 ≤ M0 < +∞, ρ0 − ρ¯ ∈ H 1 (R), v0 ∈ H 1 (R) ∩ L∞ (R). 1.1
(1.10)
Then there exists a global strong solution (ρ, v) of (1.3) on R+ × R such that for every T > 0: ρ − ρ¯ ∈ L∞ (0, T, H 1 (R)), ρ ∈ L∞ (0, T, L∞ (R)), v ∈ L∞ (0, T, H 1 (R)) ∩ L2 (0, T, H 2 (R)) and v ∈ L∞ (0, T, L∞ (R)). 5
2.5
theo
Finally this solution is unique in the class of weak solutions satisfying the usual energy inequality. Remark 1 We would like to point out that the problem remains open in the case of the 1.1 Saint-Venant system, which corresponds to system (1.3) without capillarity. In the following theorem, we are interested in proving the convergence of the global 1.1 1.3 solutions of system (1.3) to entropic solutions of the Euler system (1.5).
theo1
Theorem1.1 1.2 Let γ > 35 and (ρ , v ) with m = ρ v theo be the global solution of the Cauchy problem (1.3) with initial data (ρ0 , v0 ) as in theorem (1.1).Then, when → 0, there exists a subsequence of (ρ , m ) that converge almost everywhere to a finite entropy solution 1.3 (ρ, ρv) to the Cauchy problem (1.5) with initial data (ρ0 , ρ0 v0 ). 36
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Remark 1 We would like to point out that Lions et al in [25] had obtained the existence of global entropic solution for γ > 1 by a viscosity vanishing process, and the considered regularizing system was exactly the Korteweg system modulo the introduction of the effective velocity. theo1
One important10 basis of our problem for theorem 1.2 is the following compactness theorem established in [5]. 10
Theorem 1.3 (Chen-Perepelitsa [5]) Let ψ ∈ C02 (R), (η ψ , q ψ ) be a weak entropy pair generated by ψ. Assume that the sequences (ρ (x, t), v (x, t)) defined on R × R+ with m = ρ v , satisfies the following conditions: 1. For any −∞ < a < b < +∞ and all t > 0, it holds that: Z tZ b (ρ )γ+1 dxdτ ≤ C(t, a, b), 0
(1.11)
1.8
(1.12)
1.9
(1.13)
1.10
a
where C(t) > 0 is independent of . 2. For any compact set K ⊂ R, it holds that Z tZ (ρ )γ+θ + ρ |v |3 dxdτ ≤ C(t, K), 0
K
where C(t, K) > 0 is independent of . 3. The sequence of entropy dissipation measures −1 η ψ (ρ , m )t + q ψ (ρ , m )x are compact in Hloc (R2+ ).
Then there is a subsequence of (ρ , m ) (still denoted (ρ , m )) and a pair of measurable functions (ρ, m) such that: (ρ , m ) → (ρ, m), a.e as → 0.
(1.14)
theo2 1.9
Remark 2 We would like to recall that the estimate (1.12) was firstPer derived by Lions et 35 al in [26] by relying the moment lemma introduced by Perthame in [30]. section2
The paper is arranged as follows. In section 2 we recall some important results on the notion of entropy enrtopy-flux pair for Euler system and on the kinetic formulation of 35 section3 theo section4 Lions et al in [26]. In section 3, we show theorem 1.1 and in the last section 4.1 we prove theo1 theorem 1.2. 6
1.11
2
Mathematical tools
section2
Definition 2.1 A pair of functions (η(ρ, v), H(ρ, v)) or (η(ρ, m), q(ρ, m)) for m = ρv, 1.1 is called an entropy-entropy flux pair of system (1.3), if the following holds: [η(ρ, v)]t + [H(ρ, v)]x = 0, 1.3
for any smooth solution of (1.5). Furthermore (η(ρ, v) is called a weak entropy if: η(0, u) = 0, for any fixed v. Definition 2.2 An entropy η(ρ, m) is convex if the Hessian ∇2 η(ρ, m) is nonnegative definite in the region under consideration.
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Such η satisfy the wave equation: ∂tt η = θ2 ργ−3 ∂xx η. 35
From [26], we obtain an explicit representation of any weak entropy (η, q) under the following form: Z η ψ (ρ, m) = χ(ρ, s − v)ψ(s)ds, R Z (2.15) ψ H (ρ, m) = (θs + (1 − θ)u)χ(ρ, s − v)ψ(s)ds,
1.4
R
where the kernel χ is defined as follows: χ(ρ, v) = [ρ2θ − v 2 ]λ+ , λ = and here:
3−γ 1 γ−1 > − , and θ = , 2(γ − 1) 2 2
tλ+ =tλ for t > 0, =0 for t ≤ 0, 35
Proposition 2.1 (see [26]) For instance, when ψ(s) = 12 s2 , the entropy pair is the mechanical energy and the associated flux: m2 m3 0 η ∗ (ρ, m) = + e(ρ), q ∗ (ρ, m) = 2 + e (ρ), (2.16) 2ρ 2ρ where e(ρ) =
κ γ γ−1 ρ
1.5
represents the gas internal energy in physics.
In the sequel we will work far away of the vacuum that it why we shall introduce equilibrium states such that we avoid the vacuum. Let (¯ ρ(x), v¯(x)) be a pair of smooth −,+ ) when − + x ≥ L for some monotone functions satisfying (¯ ρ(x), v¯(x)) = (ρ−,+ , v 0 1.1 large L0 > 0. The total mechanical energy for (1.3) in R with respect to the pair of reference function (¯ ρ(x), v¯(x)) is: Z 1 ρ(t, x)|v(t, x) − v¯(x)|2 + e∗ (ρ(t, x), ρ¯(x)) dx (2.17) E[ρ, v](t) = R 2 7
1.7
3system
0
where e∗ (ρ, ρ¯) = e(ρ)e (¯ ρ) − e (¯ ρ(ρ − ρ¯) ≥ 0. The total mechanical energy for system (??) κ with κ(ρ) = ρ is: Z E1 [ρ, u](t) = R
1 1 ρ(t, x)|u(t, x) − u ¯(x)|2 + e∗ (ρ(t, x), ρ¯(x)) + 2 (∂x ρ 2 )2 dx 2
(2.18)
1.7
(2.19)
1.7
1.1
and the total mechanical energy for system (1.3) is: Z 1 E2 [ρ, v](t) = ρ(t, x)|v(t, x) − v¯(x)|2 + e∗ (ρ(t, x), ρ¯(x)) dx R 2
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Definition 2.3 Let (ρ0 , v0 ) be given initial data with finite-energy with respect to the end states: (ρ± , v ± ) at infinity, and E[ρ0 , v0 ] ≤ E0 < +∞. A pair of measurable functions 1.3 (ρ, u) : R2+ → R2+ is called a finite-energy entropy solution of the Cauchy problem (1.5) if the following properties hold: 1. The total energy is bounded in time such that there exists a bounded function C(E, t), defined on R+ × R+ and continuous in t for each E ∈ R+ with for a.e t > 0: E[ρ, v](t) ≤ C(E0 , t). 2. The entropy inequality: η ψ (ρ, v)t + q ψ (ρ, v)x ≤ 0, is satisfied in the sense of distributions for all test functions ψ(s) ∈ {±1, ±s, s2 }. 3. The initial data (ρ0 , v0 ) are obtained in the sense of distributions. 1.2
10
We now give our main conditions on the initial data (1.4), which is inspired from [5]. Definition 2.4 Let (¯ ρ(x), v¯(x)) be some pair of smooth monotone functions satisfying (¯ ρ(x), v¯(x)) = (ρ−,+ , v −,+ ) when − + x ≥ L0 for some large L0 > 0. For positive constant C0 , C1 and C2 independent of , we say that the initial data (ρ0 , v0 ) satisfy the condition H if they verify the following properties: R • ρ0 > 0, R ρ0 (x)|u0 (x) − u ¯(x)| ≤ C0 < +∞, • The energy is finite: Z 1 ρ0 (x)|v0 (x) − v¯(x)|2 + e∗ (ρ0 (x), ρ¯(x)) dx ≤ C1 < +∞, R 2 • 2
Z R
|∂x ρ0 (x)|2 dx ≤ C2 < +∞. ρ0 (x)3−2α
In this section, we would like to 1.3 recall some properties on the pair of entropy for the 1.3 system (1.5). Smooth solutions of (1.5) satisfy the conservation laws: ∂t η(ρ, u) + ∂x H(ρ, u) = 0,
8
if and only if:
0
ηρρ
P (ρ) = ηuu . ρ2
(2.20)
ondes
(2.21)
initial
ondes
We supplement the equation 2.20 by giving initial conditions: η(0, u) = 0, ηρ (0, u) = ψ(u).
35
We are now going to give a sequel of proposition on the properties of η, we refer to [26] for more details. proputile
Proposition 2.2 For ρ ≥ 0, u, ω ∈ R, ondes
initial
• The fundamental solution of (2.20)-(2.21) is the solution corresponding to ηρ (0, u) = δ(u) is given by:
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χ(ρ, ω) = (ργ−1 − ω 2 )λ+ with λ = ondes
3−γ . 2(γ − 1)
(2.22)
initial
• The solution of (2.20)-(2.21) is given by: Z ψ(ξ)χ(ρ, ξ − u)dξ, η(ρ, u) =
(2.23)
entropie
(2.24)
flux
R
• η is convex in (ρ, ρu) for all ρ, u if and only if g is convex. • The entropy flux H associated with η is given by: Z γ−1 H(ρ, u) = ψ(ξ)[θξ + (1 − θ)ξ]χ(ρ, ξ − u)dξ where θ = . 2 R 35
We now give a important result on the entropy pair (see [26], lemma 4) . pair35
Proposition 2.3 Taking ψ(s) = 12 s|s|, then there exists a positive constant C > 0, depending only on γ > 1, such that the entropy pair (η ψ , H ψ ) satisfies: |η ψ (ρ, u)| ≤ (ρ|u|2 + ργ ), H ψ (ρ, u) ≥ C −1 (ρ|u|3 + ργ+θ ), for all ρ ≥ 0 and u ∈ R, ψ |ηm (ρ, u)| ≤ (ρ|u| + ρθ ),
(2.25)
2.37
ψ |ηmm (ρ, u)| ≤ Cρ−1 .
We are now going to give recent results on the entropy pair (η ψ , q ψ ) generated by ψ ∈ 10 C02 (R) (we refer to [5] for more details). propChen
Proposition 2.4 For a C 2 function ψ : R → R, compactly supported on the interval [a, b], we have: supp(η ψ ), supp(q ψ ) ⊂ {(ρ, m) = (ρ, ρu) : u + ρθ ≥ a, u − ρθ ≤ b} :
(2.26)
Furthermore, there exists a constant Cψ such that, for any ρ ≥ 0 and u ∈ R, we have: 9
3.2
• For γ ∈ (1, 3], |η ψ (ρ, m)| + |q ψ (ρ, m)| ≤ Cψ ρ.
(2.27)
3.3
|η ψ (ρ, m)| ≤ Cψ ρ, |q ψ (ρ, m)| ≤ Cψ (ρ + ρθ+1 ).
(2.28)
3.4
(2.29)
3.5
(2.30)
3.6
• For γ ∈ (3, +∞),
• If η ψ is considered as a function of (ρ, m), m = ρu then ψ ψ |ηm (ρ, m)| + |ρηmm (ρ, m)| ≤ Cψ , ψ and, if ηm is considered as a function of (ρ, u), then ψ ψ |ηm (m, u)| + |ρ1−θ ηmρ (ρ, ρu)| ≤ Cψ .
1.3
35
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We now would like to express the kinetic formulation of (1.5) introduced in ([26]). Theorem 2.4 Let (ρ, ρv) ∈ L∞ (R+ , L1 (R)) have finite energy and ρ ≥ 0, then it is an 1.3 entropy solution of (1.5) if and only if there exists a non-positive bounded measure m on R+ × R2 such that the function χ(ρ, ξ − u) satisfies: ∂t χ + ∂x [(θξ + (1 − θ)u)χ] = ∂ξξ m(t, x, ξ).
(2.31)
cinetique theo
3
Proof of theorem 1.1
section3
Sol
We would like to start with recalling an important result due to Solonnikov (see [34]). Let ρ0 the initial density such that: 0 < m0 ≤ ρ0 ≤ M0 < +∞.
(3.32)
initiald
(3.33)
visco
When the viscosity coefficient µ(ρ) satisfies: µ(ρ) ≥ c > 0 for allρ ≥ 0, we have the existence of strong solution for small time. More exactly, we have: initiald
visco
Proposition 3.5 Let (ρ0 , v0 ) satisfy (3.32) and assume that µ satisfies (1.1 3.33), then there exists T0 > 0 depending on m0 , M0 , kρ0 − ρ¯kH 1 and kv0 kH 1 such that (1.3) has a unique solution (ρ, v) on (0, T0 ) satisfying: ρ − ρ¯ ∈ L∞ (H 1 (R), ∂t ρ ∈ L2 ((0, T1 ) × R), v ∈ L2 (0, T1 , H 2 (R)), ∂t v ∈ L2 ((0, T1 ) × R) for all T1 < T0 . Remark 3 The main point in this theorem is that the time of existence T depends only 1.1 0 of the norms of ρ0 which gives us a low bounds on T0 of the system (1.3). 10
In view of this proposition, we see that if we introduce a truncated viscosity coefficient µn (ρ): 1 µn (ρ) = max(ρ, ), n then there exists approximated solutions (ρ , v ) defined for small time (0, T0 ) of the 1.1 theo n n system (1.3). In order to prove theorem 1.1 , we only have to show that (ρn , vn ) satisfies the following bounds uniformly with respect to n and T large: 0 < m0 ≤ ρn ≤ M0 < +∞, ∀t ∈ [0, T ], 1 ρn − ρ¯ ∈ L∞ T (H (R)),
vn ∈
(3.34)
1 L∞ T (H (R)).
hal-00635983, version 1 - 26 Oct 2011
36
We are going to follow the method of Lions et al in [25], indeed the main point is to prove that we can extend the notion of Riemann invariant or more precisely1.1 the kinetic cinetique 1.1 formulation of proposition 2.4 to the system (1.3). We recall that system (1.3) has the following form: ∂ ρn + ∂x (ρn vn ) − ∂xx ρn = 0, ∂t (3.35) ∂ (ρn vn ) + ∂x (ρn vn vn ) − ∂x (∂x ρn vn ) − ∂x (ρn ∂x vn ) + ∂x (a(ρn )γ = 0, ∂t and we have finally: ∂ ρn + ∂x (ρn vn ) − ∂xx ρn = 0, ∂t ∂ (ρn vn ) + ∂x (ρn vn vn ) − ∂x ∂x (ρn vn ) + ∂x (a(ρn )γ = 0, ∂t
(3.36)
1.1a
1.1b
36
Following [25]entropie and settingflux mn = ρn vn we have for any pair of entropy flux (η(ρ, u), H(ρ, u)) defined by (2.23) and (2.24) where η is a convex function of (ρn , mn ). We write η = η¯(ρn , mn ): ∂t η + ∂x H = ¯ ηρ ∂xx ρn + ¯ ηm ∂xx mn , = ∂xx η − (¯ ηρρ (∂x ρn )2 + 2¯ ηρm (∂x ρn )(∂x mn ) + η¯mm (∂x mn )2 ). Here we define µn such that: µn = η¯ρρ (∂x ρn )2 + 2¯ ηρm (∂x ρn )(∂x mn ) + η¯mm (∂x mn )2 proputile
By proposition 2.2, we can check that µn ≥ 0. We obtain then that: ∂t η(ρn , vn ) + ∂x H((ρn , vn ) − ¯ ηρ ∂xx ρn ≤ 0 in R × (0, +∞). cinetique
By applying the same method than for proving the theorem 2.4, we obtain the following kinetic formulation: ∂t χ + ∂x ([θξ + (1 − θ)vn ]χ) − ∂xx χ = ∂ξξ m ¯ n on R2 × (0, +∞),
11
(3.37)
riemann
where m ¯ n is a nonpositive bounded measure riemann on R2 × (0, +∞). Finally we recover the classical maximum principle by multiplying (3.37) by the convex functions g(ξ) = (ξ − ξ0 )+ and g(ξ) = (ξ − ξ0 )− and integrating over R2 × (0, +∞). Indeed as we have that: −C ≤ min(v0 − ρθ0 ) ≤ max(v0 + ρθ0 ) ≤ C, x
x
and that: suppξ = [v − ρθ , v + ρ[θ]. For ξ0 large enough, we can show that: suppξ0 ∩ suppχ = ∅. We have obtain then that: −C ≤ min(v0 − ρθ0 ) ≤ vn − ρθn ≤ vn + ρθn ≤ max(v0 + ρθ0 ) ≤ C.
hal-00635983, version 1 - 26 Oct 2011
x
x
In particular we obtained that ρn and vn are uniformly bounded in L∞ (0, Tn , L∞ (R)) or: |ρn (t, x)| + |vn (t, x)|) ≤ C0 ,
sup
(3.38)
imp2
x∈R,t∈(0,Tn )
theo1
4 4.1 section4
Proof of theorem 1.2 1.1
Uniform estimates for the solutions of (1.3) 1.1
First we assume thattheo (ρ , v ) is the global solutions of Korteweg’s equations (1.3) constructed in theorem 1.1 and satisfying: ρ (t, x) ≥ c (t), for some c (t) > 0,
(4.39)
2.1
(4.40)
2.2
and lim (ρ , v )(x, t) = (ρ± , u± ).
x→±∞
Here we are working around a non constant state (¯ ρ, v¯) with: lim (¯ ρ, v¯)(x, t) = (ρ± , u± ).
x→±∞
theo
1.8
It1.9 is a simple1.10 extension of theorem 1.1. Our goal is now to check the properties (10 1.11), theo2 (1.12) and (1.13) in order to use the theorem 1.3 of Chen and Perepelista (see [5]) in theo1 order to prove the theorem 1.2. For simplicity, throughout this section, we denote (ρ, v) = (ρ , v ) and C > 0 denote the constant independent of . 1.1 We start with recalling the inequality energy for system ( 1.3), indeed by the introduction Hprepa of the effective velocity we obtain new entropies (see [13]). Lemma 1 Suppose that E1 [ρ0 , u0 ] ≤ E0 < +∞ for some E0 > 0 independent of . It holds that: Z tZ sup E1 [ρ, u](τ ) + ρu2x dxdτ ≤ C(t), (4.41) 0≤τ ≤t
0
12
R
2.3a
and: Z tZ
ρvx2 dxdτ +
sup E2 [ρ, v](τ ) + 0≤τ ≤t
lemma1
0
Z tZ 0
R
ργ−2 ρ2x dxdτ ≤ C(t),
(4.42)
2.3
R
where C(t) depends on E0 , t, ρ¯, and u ¯ but not on . 1.1
Proof: It suffices to writes the energy inequalities for system (1.3) and from system 1.3 (1.5). More exactly we have: Z Z d (η(ρ, m) − η(¯ ρ, ρ¯u ¯)dx + ρu2x dx = q(ρ− , m− ) − q(ρ+ , m+ ), dt R R with the entropy pair:
hal-00635983, version 1 - 26 Oct 2011
η(ρ, m) = with e(ρ) =
a γ γ−1 ρ .
m2 m3 0 + e(ρ), q(ρ, m) = 2 + me (ρ), 2ρ 2ρ
Since we have: (ρ, ρ¯) ≥ ρ(ρθ − ρ¯θ )2 , θ =
γ−1 , 2
we can classically bootstrap on the left hand-side the term q(ρ− , m− ) − q(ρ+ , m+ ). Remark 4 Since vacuum could occur in our solution, the inequality Z tZ ρu2x dxdτ ≤ C(t), 0
R
2.3
10
lemme2
in (4.42) is much weaker than the corresponding one in [5]. That is why lemma 2 will be more tricky to obtain. The following higher order integrability estimate is crucial in compactness argument. lemma1
lemme2
Lemma 2 If the conditions of lemma 1 hold, then for any −∞ < a < b < +∞ and all t > 0, it holds that: Z tZ b ργ+1 dxdτ ≤ C(t, a, b), (4.43) 0
2.21
a
where C(t) > 0 depends on E0 , a, b, γ, t, ρ¯, u ¯ but not on . Remark 5 The proof follows the same ideas than in the case of compressible NavierStokes equations when we wish to obtain a gain of integrability on the density. We refer fL2 Hu2 to [24] for more details. The proof is also inspired from Huang et al in [19]. Proof. Choose ω ∈ C0∞ (R) such that: 0 ≤ ω(x) ≤ 1, ω(x) = 1 for x ∈ [a, b], and suppω = (a − 1, b + 1). 1.1
By the momentum equation of (1.3) and by localizing, we have (P (ρ)ω)x = −(ρuvω)x + (P (ρ) + ρuv)ωx − (ρv)t ω + (ρvx ω)x − ρvx ωx . 13
(4.44)
2.22
2.22
Integrating (4.44) with respect to spatial variable over (−∞, x), we obtain: Z x Z x [(ρuv + P (ρ))ωx − ρvx ωx . (4.45) ρv ωdy)t + P (ρ)ω = −ρuvω + (ρvx ω)x − (
2.23
−∞
−∞
2.23
Multiplying (4.45) by ρω, we have 2
2
2
2
x
Z
2
ρP (ρ)ω = − ρ uvω + ρ vx ω − (ρω ρv ωdy)t −∞ Z x Z x [(ρuv + P (ρ))ωx − ρvx ωx ]dx, ρu ωdy) + ρω − (ρu)x ω( −∞ −∞ Z x Z x 2 2 ρv ωdy)x ρv ωdy)t − (ρuω =ρ vx ω − (ρω −∞ −∞ Z x Z x [(ρuv + P (ρ))ωx − ρvx ωx ]dx, ρv ωdy + ρω + ρuωx
2.24
(4.47)
2.25
(4.48)
2.26
(4.49)
2.27
−∞
−∞
hal-00635983, version 1 - 26 Oct 2011
(4.46)
2.24
We now integrate (4.46) over (0, t) × R and we get: Z tZ aρ 0
Z tZ
γ+1 2
0
+
Z
R
Z
−∞
Z
Z tZ
Z
R
ρv ωdy dxdτ
−∞
R
x
[(ρuv + P (ρ))ωx − ρvx ωx ]dx dxdτ.
ρω 0
ρv ωdy)dx
x
ρuωx 0
x
−∞
R
ρ0 v0 ωdy)dx + +
Z (ρω
Z tZ
x
(ρ0 ω R
2
ρ vx ω −
ω dxdτ =
R
Z
2
−∞
Let A = {x : ρ(t, x) ≥ ρ}, where ρ = 2 max(ρ+, ρ−), 2.3
then we have the following estimates by (4.42): |A| ≤
C(t) e∗ (2ρ, ρ¯)
= d(t).
2.26
By (4.48), for any (t, x) there exists a point x0 = x0 (t, x) such that |x − x0 | ≤ d(t) and ρ(t, x0 ) = ρ. Here we choose β = γ+1 2 > 0, suppx∈supp(ω) ρβ (t, x) ≤ ρβ + suppx∈supp(ω)∩A ρβ (t, x), ≤ 2ρβ + suppx∈supp(ω)∩A |ρβ (t, x) − ρβ (t, x0 )|, Z x0 +d(t) β ≤ 2ρ + suppx∈supp(ω)∩A |β||ρβ−1 (t, x)ρx |dx, x0 −d(t)
β
Z
b+1+2d(t) 2β−1
≤ 2ρ +
|β|ρ a−1−2d(t)
Z
dx + R
b+1+2d(t)
≤ C(t) +
Z
ργ dx,
a−1−2d(t)
≤ C(t). 14
2 ρ−1 ρ2x dx,
(4.50)
2.28
2.28
Using (4.50), 2.25 Young inequalities and H¨older’s inequalities, the first term on the right hand side of (4.47) is treated as follows: Z tZ ρ2 vx ω 2 dxdτ 0 R Z tZ Z tZ 1 1 ≤ ρ3 ω 4 dxdτ + ρv 2 dxdτ, 2 0 R 2 0 R x Z tZ (4.51) ρ3 ω 2 dxdτ, ≤ C(t) + 0 R Z tZ ≤ C(t) + C(t) ρ4−β ω 2 dxdτ, 0 R Z tZ ργ+1 ω 2 dxdτ, ≤ C(t) + δ 0
R
lemma1
5 3.
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2.29
Here we have used the fact that γ > By lemma 1 and the H¨older inequality, we obtain Z Z x |ρv|dy, ρvωdy ≤ supp(ω) −∞ Z Z (4.52) 1 1 2 2 2 ≤( ρdy) ( ρv dy) ≤ C(t). supp(ω)
2.30
supp(ω)
Then:
Z
Z
x
ρvωdy dx +
ρω R
Z
Z
−∞
ρ0 v0 ωdy dx
−∞
R
+
x
ρ0 ω Z tZ
Z
x
ρuωx 0
(4.53)
2.32
(4.54)
2.33
(4.55)
2.34
ρvωdy dxdτ ≤ C(t).
−∞
R
Similarly, we have:
Z tZ
Z
x
(ρuv + P (ρ))ωx dy dxdτ ≤ C(t),
ρω 0
and
−∞
R
Z tZ
Z
x
ρvx ωx dy dxdτ
ρω 0
≤
Z
ρ|vx | |ωx |dy dxdτ , R R Z Z Z 2 ρωdx ρvx dy + ρωx2 dy dτ , ρω
0
≤
−∞
R
Z tZ t
Z 0
R
R
R
≤ C(t). 2.29
2.32
2.34
2.25
Substituting (4.51), (4.53)-(4.55) into (4.47) and noticing the smallness of δ, we proved lemme2 lemma 2. lemma1
Lemma 3 Suppose that (ρ0 (x), v0 (x) satisfy the conditions in the lemmas 1. Furthermore there exists M0 > 0 independent of , such that Z ρ0 (x)|v0 (x) − v¯(x)|dx ≤ M0 < +∞, (4.56) R
15
2.35
then for any compact set K ⊂ R, it holds that: Z tZ (ργ+θ + ρ|v|3 )dxdτ ≤ C(t, K), 0
(4.57)
2.36
K
where C(t, K) is independent of . 2.36
Remark 6 In order to prove the inequality (4.57), we will use the same ingredients than 35 in [26] where this inequality was obtained for the first time. pair35
hal-00635983, version 1 - 26 Oct 2011
ψ Proof. We are now working with the function ψ of proposition 2.3. If we consider ηm as a function depending of (ρ, v), we have for all ρ ≥ 0 and v ∈ R: ( ψ |ηmv (ρ, v)| ≤ C, (4.58) ψ |ηmρ (ρ, v)| ≤ Cρθ−1 .
2.38
For this weak entropy pair (η ψ , H ψ ), we observe that: ψ
η (ρ, 0) =
ηρψ (ρ, 0)
θ = 0, H (ρ, 0) = ρ3θ+1 2 ψ
and: ψ ηm (ρ, 0)
Z
θ
= βρ with β =
Z
|s|3 [1 − s2 ]λ+ ,
R
|s|[1 − s2 ]λ+ ds.
R
By Taylor formula, we have: η ∗ (ρ, m) = βρθ m + r(ρ, m),
(4.59)
2.39
r(ρ, m) ≤ Cρv 2 ,
(4.60)
2.40
(4.61)
2.41
(4.62)
2.42
(4.63)
2.43
with: b such that, for some constant C > 0. Now we introduce a new entropy pair (b η , H) b m) = H ψ (ρ, m − ρv − ) + v − η ψ (ρ, m − ρv − ), ηb(ρ, m) = η ψ (ρ, m − ρv − ), H(ρ, with m = ρv which satisfies: ( ηb(ρ, m) = βρθ+1 (v − v − ) + r(ρ, ρ(v − v − )), r(ρ, ρ(v − v − )) ≤ Cρ(v − v − )2 .
1.1
1.1
Integrating (1.3)1 × ηbρ + (1.3)2 × ηbm over (0, t) × (−∞, x), we have: Z x Z t ηb(ρ, m) − ηb(ρ0 , m0 ) dy + q ∗ (ρ, ρ(v − v − )) + v − η ∗ (ρ, ρ(v − v − ))dτ −∞
0
∗
−
Z
= tq (ρ , 0) + 0
t
Z tZ ηbm ρvx dτ −
0
x
−∞
(b ηmu ρvx2
+ ηbmρ ρρx vx )dydτ.
2.38
By using (4.58), we obtain: Z tZ x Z tZ ηbmu ρvx2 dydτ ≤ C ρvx2 dy dτ ≤ C(t), 0
−∞
0
16
R
Z tZ
x
ηbmρ ρρx vx dydτ ≤ C
−∞
0
Z tZ ≤ C 0
2.43
0
ρvx2 dy dτ
ρθ−1 ρ|ρx vx |dy dτ ≤ C(t),
R
Z tZ + C 0
R
2.44
Z tZ
(4.64) ργ−2 ρ2x dy dτ
≤ C(t).
R
2.42
2.37
Substituting (4.63) and (4.64) into (4.62), then integrating over K and using (2.25), we obtain: Z tZ ρθ+γ + ρ|v − v − |3 dxdτ 0 K Z tZ Z tZ ∗ − ρ|v||vx |dxdτ |η (ρ, ρ(v − v )|dxdτ + C ≤ C(t) + C (4.65) 0 K 0 K Z tZ Z Z x 1+θ c ρ |vx |dxdτ + 2 sup ( eta(ρ(y, τ ), (ρv)(y, τ ))dy)dx . + C 0
τ ∈[0,t]
K
2.44
2.45
−∞
K
hal-00635983, version 1 - 26 Oct 2011
lemma1
Applying lemma 1, we have: Z tZ
|η ∗ (ρ, ρ(v − v − )|dxdτ ≤ C(t).
(4.66)
2.46
(4.67)
2.47
(4.68)
2.48
Now we are going to deal with the last term on the right hand side of (4.65). (1.3) implies that: (ρv − ρv − )t + (ρv 2 + P (ρ) − ρuu− )x = (ρvx )x . (4.69)
2.49
0
K
2.28
By H¨older’s inequality and (4.50), we get: Z tZ
1+θ
ρ 0
Z tZ |vx |dxdτ ≤ C
K
0
ρvx2 dxdτ
Z tZ 0
Z tZ
K
ρθ dxdτ,
≤ C(t) + C(t) 0
ρ1+2θ dxdτ,
+ C
K
K
≤ C(t). We have now: Z tZ
Z tZ Z tZ 1 1 ρ|v||vx |dxdτ ≤ ρvx2 dxdτ + ρv 2 dxdτ, 2 2 K 0 K 0 K ≤ C(t).
0
2.45
1.1
2.49
Integrating (4.69) over [0, t] × (−∞, x) for x ∈ K, we get: Z
x
−
Z
x
ρ(v − ρv )dy = −∞
Z
−
ρ0 (v0 − ρv )dy − −∞
t
(ρv 2 + P (ρ) − ρuu− − P (ρ− ))
0
Z +
ρvx dτ. 0
17
(4.70)
t
2.50
Furthermore: Z x ηb((ρ(y, τ ), (ρv)(y, τ ))dy −∞ Z x Z x βρθ+1 (v − v¯))dy ≤ (b η (ρρv) − βρθ+1 (v − v¯))dy + | −∞ Z−∞ Z x x (r(ρρ(v − v¯))dy + β(ρθ − (ρ− )θ )ρ(v − v¯))dy ≤ −∞ −∞ Z x ρ(v − v¯))dy , + β(ρ− )θ −∞ Z x ρ(v − v¯))dy . ≤ C(t) + β(ρ− )θ
(4.71)
2.51
−∞
2.35
lemma1 lemme2 2.50
2.51
hal-00635983, version 1 - 26 Oct 2011
By using (4.56), lemma 1 and 2, (4.70) and (4.71) we conclude the proof of the lemma.
4.2
−1 Hloc (R2+ ) Compactness
In this section we are going to take profit of the uniform estimates obtained in the previous −1 (R2+ )-compactness of section in order to prove the following lemma, which gives the Hloc the Korteweg solution sequence (ρ , v ) on a entropy- entropy flux pair. Lemma 4 Let ψ ∈ C02 (R), η ψ , H ψ ) be a weak entropy 1.1 pair generated by ψ. Then for the solutions (ρ , v ) with m = ρ v of Korteweg system (1.3, the following sequence: −1 (R2+ ) η ψ (ρ , m )t + q ψ (ρ , m )x are compact in Hloc
(4.72)
3.1
lemme4 1.1
Proof: Now we are going to prove the lemma. A direct computation on (1.3)1 × 1.1 ψ ηρψ (ρ , m ) + (1.3)2 × ηm (ρ , m ) gives: ψ ηρψ (ρ , m )t + Hρψ (ρ , m )x = ηρψ (ρ , m )(ρ )vx − ηmu (ρ , m )(ρ )(vx )2 (4.73) ψ − ηmu (ρ , m )(ρ )vx ρx .
3.7
propChen 3.6
Let K ⊂ R be compact, using proposition 2.4 (2.30) and H¨older inequality, we get: Z tZ ψ ψ |ηmu (ρ , m )(ρ )|(vx )2 + |ηmu (ρ , m )(ρ )vx ρx |dxdt 0 K Z tZ Z tZ (4.74) (ρ )|(vx )2 dxdτ + C (ρ )γ−2 (ρx )2 dxdτ ≤ C 0
K
0
3.8
K
≤ C(t). This shows that: ψ ψ −ηmu (ρ , m )ρ (vx )2 − ηmu (ρ , m )ρ vx ρx are bounded in L1 ([0, T ] × K), −1,p1 and thus it is compact in Wloc (R2+ ), for 1 < p1 < 2. Moreover we observe that ψ |ηmu (ρ , ρ v )| ≤ Cψ ,
18
(4.75)
3.9
, then we obtain: Z tZ
4
ψ (ηm (ρ , m )ρ vx ) 3 dxdt 0 K Z tZ 4 4 4 3 |ρ | 3 |vx | 3 dxdt ≤ 0 K Z tZ Z tZ 4 4 2 ρ |vx | dxdt + C 3 (ρ )2 dxdt ≤ C 3 0 K 0 K Z tZ 1 4 (ρ )γ+1 dxdt →→0 0. ≤ C(t, K) 3 + C 3 0
3.10
(4.76)
3.10
(4.77)
3.11
(4.78)
3.12
K
3.9
Using (4.76) and (4.75), we obtain −1,p1 (R2+ ) for some 1 < p1 < 2. ηρψ (ρ , m )t + Hρψ (ρ , m )x are compact in Wloc
3.3
3.4
lemma1 lemme2 2.36
hal-00635983, version 1 - 26 Oct 2011
Furthermore by (2.27)-(2.28), lemma 1-2 and (4.57), we have: 3 ηρψ (ρ , m )t + Hρψ (ρ , m )x are uniformly bounded in Lploc (R2+ ) for p3 > 2.
where p3 = γ + 1 > 2 when γ ∈ (1, 3], and p = lemme4 3 we conclude the proof of the lemma 4.
γ+θ 1+θ
> 2 when γ > 3. By interpolation
theo1
5
Proof of theorem 1.2 lemma1
theo2
From lemmas 1, we have verified the theo2 conditions (i)-(iii) of theorem 1.3 for the sequence of solutions (ρ , m ). Using theorem 1.3, there exists a subsequence (ρ , m ) and a pair of measurable functions (ρ, m) such that (ρ , m ) → (ρ, m), a.e → 0.
(5.79)
It is easy 1.3 to check that (ρ, m) is a finite-energy entropy solution (ρ, m) to the Cauchy 5 problem (1.5) with initial data (ρtheo1 0 , ρ0 u0 ) for the isentropic Euler equations with γ > 3 . It achieves the proof of theorem 1.2.
References fA
[1] D.M. Anderson, G.B McFadden and A.A. Wheller. Diffuse-interface methods in fluid mech. In Annal review of fluid mechanics, Vol. 30, pages 139-165. Annual Reviews, Palo Alto, CA, 1998.
BB1
[2] S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math., 161 1 (2005), pp. 223342.
fC
[3] J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system, I. Interfacial free energy, J. Chem. Phys. 28 (1998) 258-267.
Chen
[4] G. Q. Chen, The theory of compensated compactness and the system of isentropic gas dynamics, Lecture notes, Preprint MSRI-00527-91, Berkeley, October 1990. 19
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[5] G. Chen and M. Perepelista, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow, Comm. Pure. Appl. Math. 2010 (to appear).
fDD
[6] R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Annales de l’IHP, Analyse non lin´eaire 18,97-133 (2001).
Di1
[7] R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Commun. Math. Phys. 91 (1983), 1-30.
Di2
[8] R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal. (1983), 22-70.
fDS
[9] J.E. Dunn and J. Serrin, On the thermomechanics of interstitial working , Arch. Rational Mech. Anal. 88(2) (1985) 95-133.
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Evans
[10] L. C. Evans, Partial differential equations, AMS, GSM Volume 19.
G
[11] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,.Comm. Pure Appl. Math. 18. 1965 (697-715).
fGP
[12] M.E. Gurtin, D. Poligone and J. Vinals, Two-phases binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci. 6(6) (1996) 815–831.
Hprepa
[13] B. Haspot, Blow-up criterion, ill-posedness and existence of strong solution for Korteweg system with infinite energy, preprint (arxiv February 2011).
Hprepa1
[14] B. Haspot, New entropy for Kortewegs system, existence of global weak solution and Prodi-Serrin theorem, preprint (arxiv February 2011).
fH1
[15] B. Haspot, Existence of solutions for compressible fluid models of Korteweg type, Annales Math´ematiques Blaise Pascal 16, 431-481 (2009).
fH2
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